You may have to use the 1/4 inch square mesh/hardware cloth. I've had this happen and it could only have been a rat where I live. Rats can climb wood very easily and if your quail was sleeping too close to the gaps in the wire the rat just grabs it through the wire. Traps can take awhile to catch the culprit and where there's one, there's always more.
I've also had to put plastic sheeting and insect mesh on my aviary (which has 1/4 inch mesh) after losing finch chicks and eggs to mice. I witnessed a mouse squeezing through the 1/4 inch gaps to escape after I caught it in there. I've been raising 4 finch chicks since they were a day old as they were attacked and they all have missing toes, one quite severely (I was very surprised he survived). The mice would've just kept coming back if I'd left them in the aviary. I naively thought mice just ate grains and seeds!
I've also had to put plastic sheeting and insect mesh on my aviary (which has 1/4 inch mesh) after losing finch chicks and eggs to mice. I witnessed a mouse squeezing through the 1/4 inch gaps to escape after I caught it in there. I've been raising 4 finch chicks since they were a day old as they were attacked and they all have missing toes, one quite severely (I was very surprised he survived). The mice would've just kept coming back if I'd left them in the aviary. I naively thought mice just ate grains and seeds!